Concentration of positive solutions for a class of fractionalp-Kirchhoff type equations

We study the existence and concentration of positive solutions for the following class of fractional p -Kirchhoff type problems: $$ \{{array}{@{}ll} (εˢᵖa+ε²ˢᵖ⁻³b \,[u]s, pᵖ)(-Δ)ₚˢu+V(x)uᵖ⁻¹=f(u) & in\ R³,\\ { u∈ Ws, p(R³), u>0 & in\ R³, {array}.$$ where ɛ is a small positive parameter, a and b are positive constants, s ∈ (0, 1) and p ∈ (1, ∞) are such that $sp ∈ (3/2, 3)$ , $(-Δ )ˢₚ$ is the fractional p -Laplacian operator, f : ℝ → ℝ is a superlinear continuous function with subcritical growth and V : ℝ 3 → ℝ is a continuous potential having a local minimum. We also prove a multiplicity result and relate the number of positive solutions with the topology of the set where the potential V attains its minimum values. Finally, we obtain an existence result when f ( u ) = u q −1 + γ u r −1 , where γ > 0 is sufficiently small, and the powers q and r satisfy 2 p < q < p * s ⩽ r . The main results are obtained by using some appropriate variational arguments.